Abstract: In the classical portfolio selection theory the value of the assets is considered infinitely divisible. In the real portfolio selection models one should consider only finitely divisible assets. This is because the investors purchase only a finite number of shares or minimum transaction lots. We present several asset allocation models in discrete variable and we make an analysis of the results. Our models are closer to reality but they are more difficult to be solved.

Dr. Marius Rădulescu graduated Faculty of Mathematics in 1977. He took his Ph.D. degree in 1985 at Centre of Mathematical Statistics “Gheorghe Mihoc” in Bucharest. In 1991 he was awarded a prize of the Romanian Academy. At present he is a senior research worker at Institute of Mathematical Statistics and Applied Mathematics Gheorghe Mihoc-Caius Iacob in Bucharest. He published several books and research papers in the area: nonlinear functional analysis and its applications to boundary value problems for differential equations, real analysis, numerical analysis, optimization theory, approximation theory, mathematical modeling (operations research), portfolio theory. He is a member of the Editorial Board of Journal of Applied Sciences, Advanced Modeling and Optimization and Arhimede.

Dr. Constanţa Zoie Rădulescu graduated Faculty of Mathematics in 1977. She is a senior research worker at National Institute for Research and Development in Informatics in Bucharest. She has got a Ph.D. degree at Centre of Mathematical Statistics in Bucharest. Her scientific interests are: DSS for the management of financial investments, multicriteria decision analysis, risk analysis, mathematical modeling (operations research).

Prof. dr. Gheorghiţă Zbăganugraduated Faculty of Mathematics in 1975. He took his Ph.D. degree in 1986 at Centre of Mathematical Statistics “Gheorghe Mihoc” in Bucharest. In 1999 he was awarded a prize of the Romanian Academy. At present he is a senior research worker at Institute of Mathematical Statistics and Applied Mathematics Gheorghe Mihoc – Caius Iacob in Bucharest and a professor at the Faculty of Mathematics and Computer Science, University of Bucharest He has published several books and research papers in the area: probability theory, ruin theory, actuarial science, information theory, combinatorics, optimization theory, functional analysis.

The original Markowitz model of portfolio selection has received a widespread theoretical acceptance and it has been the basis for various portfolio selection techniques. The model is known in the literature also, as the mean-variance portfolio selection model. Generally, in the classical mean-variance portfolio selection approach several realistic features are not taken into account.

Among these “forgotten” aspects, one of particular interest is the not infinite divisibility of the financial asset to select, i.e. the obligation to buy/sell only integer quantities of asset lots which contain a predetermined number of shares.

The portfolio selection problems with discrete constraints (such as buy-in thresholds, cardinality constraints and transaction roundlot restrictions) were studied in the literature mainly in the last decade. Integer programming approaches in the mean-risk models were studied in [3]-[14], [16], [17]. Mansini and Speranza in [9] consider the constraint stating that assets can be traded only in indivisible lots of fixed size. In this case, the problem is formulated in terms of integer-valued variables – as opposed to real-valued ones – that represent, for each asset, the number of purchased lots, instead of the real-valued ones.

Given that assets are normally composed by units, this constraint is certainly meaningful; its practical importance however depends on the ratio between the size of the minimum trading lots and the size of the shares involved in the portfolio. In [4] Corazza and Favaretto study the existence problem for the solutions of a discrete mean-variance portfolio selection model.

Mansini and Speranza in [10] consider a single-period mean-safety portfolio selection problem with transaction costs and integer constraints on the quantities selected for the securities (rounds). They propose an exact approach based on the partition of the initial problem into two sub-problems and the use of a simple local search heuristic to obtain an initial solution.

In [6], [9], [11] and [17] are presented heuristic algorithms for the portfolio selection problem with minimum transaction lots.

In [3] is presented an approach of the mean-variance portfolio selection with the nonlinear mixed-integer programming. The authors proposed an algorithm (which is based on the proposed conditions) for finding a “good” feasible solution and proved its convergence.

In [8] the classic mean-variance framework is extend to a broad class of investment decisions under risk where investors select optimal portfolios of risky assets that include perfectly divisible as well as perfectly indivisible assets. The author develops an algorithm for solving the associated mixed-integer nonlinear program and report on the results of a computational study.

In [5] are examined the effects of applying buy-in thresholds, cardinality constraints and transaction roundlot restrictions to the portfolio selection problem. Such discrete constraints are of practical importance but make the efficient frontier discontinuous. The resulting quadratic mixed-integer (QMIP) problems are NP-hard and therefore computing the entire efficient frontier is computationally challenging. The authors proposed alternative approaches for computing this frontier and provided insight into its discontinuous structure.

A fundamental question in the mathematical finance is how risk should be measured properly. The mean-variance models behave well in the case when the distribution of the random vector of returns is close to a multivariate normal distribution, and not so well in other cases. An important problem is to build portfolio selection models based on risk measures that capture risk adequately. In 1952, Roy [15] suggested that investors are interested in selecting a portfolio so as to maximize the probability of achieving at least a given return. The idea is that if return falls below the threshold there will be some bad consequence. This model of investor behavior is called safety­first. Drawing the efficient frontier in standard deviation ­ expected return space, the portfolio which maximizes the probability of realized return being greater than the threshold, can be found by identifying the straight line passing through the expected return axis at the threshold that is tangent to efficient frontier. The portfolio at the point of tangency is the desired portfolio.

The development of the theory of stochastic dominance had stimulated the research for asymmetric risk measures (downside risk measures) like: shortfall probability, expectations of loss, semi-variance and lower partial moments. An important class of risk measures considered in the literature is the coherent risk measures [1], [14].

A new tool for financial analysis is the Omega function [2]. If X is a random variable denote by the cumulative distribution function of X. The Omega function associated to X and to the interval is defined as follows:

,

The Omega function has the advantage that incorporates all the information of the returns. The evolution of the Omega function over the time provides a complete picture of performance and risk.

In order to consider the discrete feature, we build several linear integer programming problems. The present paper continues the ideas from [13]. The risk measure considered in this paper is the lower partial moment of the first order. We propose a formulation of this problem in terms of quantities, i.e. integer numbers of asset lots to buy, instead of starting capital percentages to invest. We give necessary and sufficient conditions for the existence of feasible solution(s) with a sequence of steps to be followed by the investor when he wants to make investment decisions in the presence of minimum transaction lots. A numerical example illustrating the previous points is presented.